(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids

The Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to many areas in computer science and natural sciences. In this paper, we show that for the Boolean hypercube \B^n, the randomized query complexity of Local Search is $\Theta(2^{n/2}n^{1/2})$ and the quantum query complexity is $\Theta(2^{n/3}n^{1/6})$. We also show that for the constant dimensional grid $[N^{1/d}]^d$, the randomized query complexity is $\Theta(N^{1/2})$ for $d \geq 4$ and the quantum query complexity is $\Theta(N^{1/3})$ for $d \geq 6$. New lower bounds for lower dimensional grids are also given. These improve the previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04]. Finally we show for $[N^{1/2}]^2$ a new upper bound of $O(N^{1/4}(\log\log N)^{3/2})$ on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.Comment: 18 pages, 1 figure. v2: introduction rewritten, references added. v3: a line for grant added. v4: upper bound section rewritte

Ant-Inspired Density Estimation via Random Walks

Many ant species employ distributed population density estimation in applications ranging from quorum sensing [Pra05], to task allocation [Gor99], to appraisal of enemy colony strength [Ada90]. It has been shown that ants estimate density by tracking encounter rates -- the higher the population density, the more often the ants bump into each other [Pra05,GPT93]. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides new tools for understanding complex dependencies in the collision probabilities of multiple random walks. We bound the strength of these dependencies using $local\ mixing\ properties$ of the underlying graph. Our results extend beyond the grid to more general graphs and we discuss applications to size estimation for social networks and density estimation for robot swarms

Graph ambiguity

In this paper, we propose a rigorous way to define the concept of ambiguity in the domain of graphs. In past studies, the classical definition of ambiguity has been derived starting from fuzzy set and fuzzy information theories. Our aim is to show that also in the domain of the graphs it is possible to derive a formulation able to capture the same semantic and mathematical concept. To strengthen the theoretical results, we discuss the application of the graph ambiguity concept to the graph classification setting, conceiving a new kind of inexact graph matching procedure. The results prove that the graph ambiguity concept is a characterizing and discriminative property of graphs. (C) 2013 Elsevier B.V. All rights reserved

The role of information flow in engineering optimization

Current optimization techniques work well for single components represented by a single model. However, many of the problems we face today are multi-disciplinary problems requiring the integration of complex models from different fields to gain a more complete understanding of the overall performance of a biological, engineering, or human system. One example is a modern automobile. Multiple systems (such as the power train and electronic engine control system) are designed and built from various assemblies and components, all of which are then integrated into one final product. This design process evokes a systems-of-systems concept that is also found in agricultural facilities, aircraft design, and many other industrial applications where multiple systems are orchestrated to achieve common goals. Optimization of these complex systems is challenging. Tight coupling between the various models, discontinuous search spaces, and long run times can quickly defeat traditional optimization techniques.;Evolutionary algorithms provide a way to approach optimization of these complex systems. Evolutionary algorithms blend the information contained in a population of solutions to answer problems that thwart many classical optimization methods, but loss of diversity in the evolving solutions is a critical issue. As this information is shared between the population members, the diversity in that population decreases as the solutions converge to a single answer. For many challenging engineering problems this loss of diversity occurs too rapidly for novel solutions to emerge. In addition, systems of systems optimization problems are often deceptive because the global optimum is composed of multiple building blocks, making the preservation of diversity crucial.;This work presents graph based evolutionary algorithms as a tool to control the rate at which information is spread throughout an evolving population and thereby limit diversity loss. Graph based evolutionary algorithms impose a computational geography on the evolving population, placing barriers to information flow to allow for the development of the building blocks required to assemble one or more superior solutions. Graph based evolutionary algorithms can be used to find new solutions and decrease the time to convergence to a global optimum for complex, deceptive problems. In addition, the performance of a problem on a set of graphs can be used as a taxonomical character to classify evolutionary computation problems. If comparisons can be made between classified problems and a new problem being examined, it would be possible to select a graph that matches the desired performance. This careful graph selection can provide solutions that are both novel and superior to those found by standard evolutionary algorithms. Successful examples can be found in a variety of disciplines, including the engineering design problem of optimizing cook stoves for use in the third world to biological systems-of-systems, such as the tailoring of antibiotic regimens for use in swine production